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The purpose of this article is to present a series of case studies which examined the influence of two methods of multiplication fact instruction, CTD and SIM, with regard to maintenance and generalization. Maintenance was defined the number of correct digits per minute written when presented with previously taught multiplication facts. Maintenance was measured 1 week after instruction and 5 weeks after instruction. Generalization was defined as computation of unknown multiplication facts. Generalization was measured in two ways: (a) through a timed (1-minute) probe that included a set of unknown multiplication facts, and (b) through an untimed probe that included a different set of unknown multiplication facts. This article focuses on application of CTD and SIM instruction and their effects on the performance of students with specific learning disabilities in the area of mathematics.
Setting and Participation Requirements
Participants were students with specific learning disabilities attending sixth grade in a major southeastern city. The school was chosen based a recommendation by the district administration. Students were selected based on their area of disability eligibility, the presence of prerequisite math skills, and their knowledge of multiplication. Students whose area of eligibility was listening comprehension were not considered since this would interfere with probe administration and parts of instruction that were presented verbally. In order to participate in the study, students met the following criteria as indicated by Mercer and Miller (1992a) in the Strategic Math Series: (a) count to 81, (b) compute addition facts to 18, (c) add single column sums to 81 without regrouping, and (d) copy 40 digits in 1 minute. Students were also selected based on their lack of skill mastery with regard to multiplication facts. That is, students needed a pool of at least thirty unknown multiplication facts. The range of unknown multiplication facts was 30 to 50 multiplication facts. All of the students performed at least three grade levels below their grade placement in reading, math, and writing. The interventions took place during a thirty-minute period designated for remedial instruction. This period was school-wide, not a special education initiative. The primary researcher provided one-on-one instruction with each student for 15-20 minutes per session. The primary researcher was a white female certified in special education, with 7 years of experience. The instructor received professional development in both CTD and SIM through her teacher preparation program and professional development workshops. She implemented both instructional methods successfully with students with disabilities in her previous position as a middle school resource teacher.
SIM was implemented according to Mercer and Miller’s (1992a) Strategic math series: Multiplication facts 0-8. CTD was conducted according to a modified version of procedures outlined by Wolery, Ault, and Doyle (1992). All procedures were identical to Wolery et al., with the exception that students were told to write their answer instead of simply providing a verbal response. Since SIM requires writing, this was an attempt to equalize time issues across both methods.
Prior to instruction, students completed a multiplication test that included 100 multiplication facts, factors 0-9. The multiplication facts that were missed were grouped into four independent (no duplicated facts such as 6x8 and 8x6) and balanced groups of facts. To ensure that CTD did not influence SIM, groups used within each type of instruction consisted of facts that were less likely to influence the learning of other facts (for example, knowing 9x5 might lead to solving 9x6 by solving 45+9.). No strategies were taught within either method that might have encouraged such problem solving, but efforts were made to minimize the chance incidental learning. The four groups of facts were used for the two intervention phases (ten facts for each condition) and the two sets of generalization probes (five facts for each probe). Students’ writing fluency was also measured to ensure adequate motor skills to reach criterion of writing 40 digits per minute with no more than two errors. Given 1 minute, the students copied numbers written on a page.
The materials for CTD consisted of multiplication facts printed vertically in black ink on a 4-by-6 inch index card with print that was approximately 2 inches in height. The order of the flashcards was changed between each trial so that the order of presentation was not predictable. In addition to the flashcards, instructional materials included a multiplication sheet that corresponded to the flashcards. After verbally responding to the flashcards, the students wrote the answer on the multiplication sheet. This modification of CTD materials was to prepare the students for the written probes. The materials for fluency instruction consisted of a sheet containing the target facts, written multiple times in a random order.
During SIM, the materials included a contract in which the teacher and student agreed to work rigorously using strategies to master multiplication facts. The materials for lessons 1-3 were paper plates, three- dimensional objects, and a worksheet with target multiplication facts written vertically and horizontally. The materials for representational phase involved drawings that were pre-printed on the lesson sheet and target multiplication facts written vertically and horizontally. After instruction at the representational level, the DRAW (Discover the sign, Read the problem, Answer, or draw tallies and/or circles and check your answer, Write the answer) was introduced where the materials consisted of a sheet printed with the strategy. The last phase of instruction involved fluency activities and the materials included facts written repeatedly and in a random order. The materials also included a chart used to record student progress after each activity. The probes for CTD and SIM consisted of multiplication facts written repeatedly in random order.
Treatment integrity was conducted during 25% of the lessons (Poling, Methot, & LeSage, 1995). Live observations were used to evaluate the instructor’s behaviors. A treatment checklist for each intervention was used in order to ensure that procedures were carried out correctly. A teacher was trained in using the treatment integrity checklists through demonstration and practice. When the teacher completed a checklist with 100% accuracy, treatment integrity checks began. Treatment integrity was calculated at 100% for the study.
Inter-rater reliability was conducted for 25% of all of the multiplication probes administered. A teacher was trained through demonstration and practice in the scoring procedures and reliability checks began after the teacher had scored a probe with 100% accuracy. The primary researcher collected data on a daily basis and scored fluency probes. The trained teacher scored the same multiplication probes. To calculate inter-rater reliability, the total number of agreements between the teacher and the primary researcher were divided by the total number of observations and this answer was multiplied by one hundred (Poling et al., 1995). Inter-rater reliability was calculated for CTD responses, independent written work during SIM, fluency, and mastery probes. Inter-rater reliability was 100% for the study.